Corollary 2 of Euler's Theorem Formula and Proof

TL;DR

The corollary states that if there is a function u of degree n and another function z which is homogeneous of degree n, then the equation x(d u/d x) + y(d u/d y) = n(f(u)/f'(u)) holds.

Transcript

hello in this session we'll see corollary 2 of euler's theorem we'll see the formula and it's proof so in this corollary 2 let's say we have a function u in x and y with degree n but u is or you may not be a homogeneous function so in that case if there is another function let's say z which is function of u and this is homogeneous of degree n then ... Read More

Key Insights

• 🚱 Corollary 2 of Euler's theorem explains the relationship between non-homogeneous and homogeneous functions.
• 😄 The equation x(d u/d x) + y(d u/d y) = n(f(u)/f'(u)) provides a way to calculate derivatives and understand the behavior of such functions.
• 📏 The proof involves substituting the homogeneous function with f(u) and applying the chain rule to find the derivatives.
• 🏑 This corollary has applications in various fields, including physics, economics, and engineering.
• 👻 The corollary allows for the analysis of relationships and properties of functions in both theoretical and real-world scenarios.
• ❓ Euler's theorem is a fundamental result in mathematics that has implications in different areas.
• 🚱 Understanding the relationship between homogeneous and non-homogeneous functions helps in solving differential equations and optimization problems.

Questions & Answers

Q: What does corollary 2 of Euler's theorem state?

Corollary 2 states that if we have a non-homogeneous function u of degree n and a homogeneous function z of degree n, then x(d u/d x) + y(d u/d y) = n(f(u)/f'(u)).

Q: How is the proof of the corollary carried out?

The proof involves considering z as a function of u, applying Euler's theorem to z, substituting it with f(u), and using the chain rule to differentiate and simplify the equation until we obtain x(d u/d x) + y(d u/d y) = n(f(u)/f'(u)).

Q: What is the significance of this corollary?

Corollary 2 provides a mathematical relationship between non-homogeneous and homogeneous functions, allowing for the calculation of derivatives and understanding the behavior of such functions.

Q: Can this corollary be applied in real-world scenarios?

Yes, this corollary can be applied in various fields such as physics, economics, and engineering, where non-homogeneous and homogeneous functions are used to model real-world phenomena. It helps in analyzing the relationships between variables and understanding their properties.

Summary & Key Takeaways

• Corollary 2 of Euler's theorem explains the relationship between a function u and a function z, where z is a homogeneous function of the degree n that depends on u.

• The corollary states that the equation x(d u/d x) + y(d u/d y) = n(f(u)/f'(u)) holds.

• The proof of the corollary involves applying Euler's theorem to the function z, substituting it with f(u), and using the chain rule to find the differentiation.